张量几何：第2版

Introduction

0.FundamentalNot(at)ions

1.Sets

2.Functions

3.PhysicalBackground

Ⅰ.RealVectorSpaces

1.Spaces

Subspacegeometry,components

2.Maps

Linearity,singularity,matrices

3.Operators

Projections,eigenvMues,determinant,trace

Ⅱ.AffineSpaces

1.Spaces

Tangentvectors,parallelism,coordinates

2.CombinationsofPoints

Midpoints,convexity

3.Maps

Linearparts,translations,components

Ⅲ.DualSpaces

1.Contours,Co-andContravariance,DualBasis

Ⅳ.MetricVectorSpaces

1.Metrics

Basicgeometryandexamples,Lorentzgeometry

2.Maps

Isometries,orthogonalprojectionsandcomplements,adjoints

3.Coordinates

Orthonormalbases

4.DiagonalisingSymmetricOperators

Principaldirections,isotropy

Ⅴ.TensorsandMultilinearForms

1.MultilinearForms

TensorProducts,Degree,Contraction,RaisingIndices

Ⅵ.TopologicalVectorSpaces

1.Continuity

Metrics:topologies,homeomorphisms

2.Limits

Convergenceandcontinuity

3.TheUsualTopology

Continuityinfinitedimensions

4.CompactnessandCompleteness

IntermediateValueTheorem,convergence,extrema

Ⅶ.DifferentiationandManifolds

1.Differentiation

Derivativeaslocallinearapproxiamation

2.Manifolds

Charts,maps,diffeomorphisms

3.BundlesandFields

Tangentandtensorbundles,metrictensors

4.Components

HairyBallTheorem,transformationformulae,raisingindic

5.Curves

Parametrisation,length,integration

6.VectorFieldsandFlows

Firstorderordinarydifferentialequations

7.LieBrackets

Commutingvectorfieldsandflows

Ⅷ.ConnectionsandCovariantDifferentiation

1.CurvesandTangentVectors

Representingavectorbyacurve

2.RollingWithoutTurning

Differentiationalongcurvesinembeddedmanifolds

3.DifferentiatingSections

Connectionshorizontalvectors,Christoffelsymbols

4.ParallelTransport

Integratingaconnection

5.TorsionandSymmetry

Torsiontensorofaconnection

6.MetricTensorsandConnections

Levi-Civitaconnection

7.CovariantDifferentiationofTensors

Paralleltransport,RiccisLemma,components,constancy

Ⅸ.Geodesics

1.LocalCharacterisation

Undeviatingcurves

2.GeodesicsfromaPoint

Completeness,exponentialmap,normalcoordinates

3.GlobalCharacterisation

Criticalityoflengthandenergy,FirstVariationFormula

4.Maxima,Minima,Uniqueness

Saddlepoints,mirages,TwinsParadox

5.GeodesicsinEmbeddedManifolds

Characterisation,examples

6.AnExampleofLieGroupGeometry

2x2matricesasapseudo-Riemannianmanifold

Ⅹ.Curvature

1.FlatSpaces

Intrinsicdescriptionoflocalflatness

2.TheCurvatureTensor

PropertiesandComponents

3.CurvedSurfaces

Ganssiancurvature,Gauss-BonnetTheorem

4.GeodesicDeviation

Tidaleffectsinspacetime

5.SectionalCurvature

SchursTheorem,constantcurvature

6.RicciandEinsteinTensors

Signs,geometry,Einsteinmanifolds,conservationequation

7.TheWeylTensor

Ⅺ.SpecialRelativity

1.OrientingSpacetimes

Causality,particlehistories

2.MotioninFlatSpacetime

Inertialframes,momentum,restmass,mass-energy

3.Fields

Mattertensor,conservation

4.Forces

Noscalarpotentials

5.GravitationalRedShiftandCurvature

Measurementgivesacurvedmetrictensor

Ⅻ.GeneralRelativity

1.HowGeometryGovernsMatter

Equivalenceprinciple,freefall

2.WhatMatterdoestoGeometry

Einsteinsequation,shapeofspacetime

3.TheStarsinTheirCourses

Geometryofthesolarsystem,Schwarzschildsolution

4.FarewellParticle

Appendix.ExistenceandSmoothnessofFlows

1.Completeness

2.TwoFixedPointTheorems

3.SequencesofFunctions

4.IntegratingVectorQuantities

5.TheMainProof

6.InverseFunctionTheorem

Bibliography

IndexofNotations

Index

　　《张量几何(第2版)(英文版)》是Springer数学研究生丛书之一，是一部详细讲述张量几何的教程。书中对微分几何的处理方式，以及学习广义相对论需要的数学知识使得本教程对于稍微了解单变量基本微积分和一些向量代数的知识就可以完全读懂该书的内容。《张量几何(第2版)(英文版)》用以书的形式能够提供的三维或更多维的图的形式使得内容更加形象化，重点强调数学的几何。为了表达的流畅和增强可读性，许多证明都是以练习的形式展示给读者，而非长篇的列举方程。这样，读者只能亲自进行实际计算，而不是跳过现成的例子。　　这本内容丰富的教程对微分几何在相对论研究中的应用是个巨大的贡献。